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Theorem dmv 5760
 Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
Assertion
Ref Expression
dmv dom V = V

Proof of Theorem dmv
StepHypRef Expression
1 ssv 3942 . 2 dom V ⊆ V
2 dmi 5759 . . 3 dom I = V
3 ssv 3942 . . . 4 I ⊆ V
4 dmss 5739 . . . 4 ( I ⊆ V → dom I ⊆ dom V)
53, 4ax-mp 5 . . 3 dom I ⊆ dom V
62, 5eqsstrri 3953 . 2 V ⊆ dom V
71, 6eqssi 3934 1 dom V = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  Vcvv 3444   ⊆ wss 3884   I cid 5427  dom cdm 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-dm 5533 This theorem is referenced by: (None)
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